Fundamental Groups of Random Clique Complexes

Abstract

Clique complexes of Erdős-Rényi random graphs with edge probability between $n^{-\frac{1}{3}}$ and $n^{-\frac{1}{2}}$ are shown to be aas not simply connected. This entails showing that a connected two dimensional simplicial complex for which every subcomplex has fewer than three times as many edges as vertices must have the homotopy type of a wedge of circles, two spheres and real projective planes. Note that $n^{-\frac{1}{3}}$ is a threshold for simple connectivity and $n^{-\frac{1}{2}}$ is one for vanishing first $\F_2$ homology.

7 pages statement of Theorem 1.2 corrected

References in corpus (1)

• The fundamental group of random 2-complexes